116 THE PRINCIPLES OF SCIENCE. [CHAP. 



the combinations of terms which it destroys. Hence two 

 propositions are equivalent when they remove the same 

 combinations from the Logical Alphabet, and neither more 

 nor less. A proposition is inferrible but riot equivalent to 

 another when it removes some but not all the combinations 

 which the other removes, and none except what this 

 other removes. Again, propositions are consistent .provided 

 that they jointly allow each term and the negative of 

 each term to remain somewhere in the Logical Alphabet. 

 If after all the combinations inconsistent with two propo- 

 sitions are struck out, there still appears each of the letters 

 A, a, B, b, C, c, D, d, which were there before, then no 

 inconsistency between the propositions exists, although 

 they may not be equivalent or even inferrible. Finally, 

 contradictory propositions are those which taken together 

 remove any one or more letter-terms from the Logical 

 Alphabet. 



What is true of single propositions applies also to groups 

 of propositions, however large or complicated ; that is to 

 say, one group may be equivalent, inferrible, consistent, 

 or contradictory as regards another, and we may similarly 

 compare one proposition with a group of propositions. 



To give in this place illustrations of all the four kinds 

 of relation would require much space : as the examples 

 given in previous sections or chapters may serve more or 

 less to explain the relations of inference, consistency, and 

 contradiction, I will only add a few instances of equivalent 

 propositions or groups. 



In the following list each proposition or group of pro- 

 positions is exactly equivalent in meaning to the corre- 

 sponding one in the other column, and the truth of this 

 statement may be tested by working out the combinations 

 of the alphabet, which ought to be found exactly the same 

 in the case of each pair of equivalents. 



A = Ab B = aB 



A =. BC . . .' ! .' a = b -I- c 



A = AB -I- AC . . . b = 



A.|-B = C.|.D . . . . ab = cd 



A -I- c = B -I- d . . . . aC = bV 



A AKr I A7.P f A = AB-|-AC 



A = ABc.|.AJC I AB = ABc ' 



